CHEMICAL ENGINEERING TRANSACTIONS VOL. 81, 2020 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš Copyright © 2020, AIDIC Servizi S.r.l. ISBN 978-88-95608-79-2; ISSN 2283-9216 Analysis of Factors Affecting Dropwise Condensation Heat Transfer Based on Theoretical Model Jie Pang, Jinlan Gou, Chonghai Huang, Kai Chen, Qi Xiao, Shiwei Yao* Science and Technology on Thermal Energy and Power Laboratory Wuhan, Hubei, China crab2003@qq.com Based on the processes of droplet nucleation, growth, coalescence and departure, the heat transfer model of dropwise condensation of a pure vapour on hydrophobic/superhydrophobic surfaces is developed. The influences of the number of nucleation site, minimum radius, contact angle and departure radius on heat transfer performance are studied by numerical simulation. The results indicate that the minimum radius affects the dropwise condensation a little while it is much less than the characteristic coalescence radius. The internal droplet thermal resistance increases as the contact angle increases. As a result, the heat transfer performance of superhydrophobic surfaces is not as good as that of hydrophobic surfaces. 1. Introduction There are two forms of vapour condensation, i.e. filmwise condensation (FWC) and dropwise condensation (DWC). Schmidt et al. (1930) found dropwise condensation of which the heat transfer coefficient was several times or even dozens of times of filmwise condensation (Rose, 2002). Since then, dropwise condensation had been paid much more attention to scientific research and industrial application. Dropwise condensation contains the lifecycle of droplet nucleation, growth, coalescence and departure. The behaviour characteristic of droplets is the basis for understanding the law, build the physical model and predict the performance of dropwise condensation heat transfer. Le Fevre and Rose (1966) first proposed a theoretical model of dropwise condensation heat transfer and used the droplet size distribution function fitted from experimental data. Based on assuming that the droplet size distribution stays the same, Wu and Maa (1976) built the population balance model and obtained the droplet size distribution of small droplets which have grown by direct condensation. Abu-Orabi (1998) combined the population balance model and droplet size distribution function, given a theoretical model of hemispherical droplets on a vertical surface. Kim and Kim (2011) extended Abu-Orabi’s model to inclined superhydrophobic surfaces and derived the thermal resistance inside the droplet analytically. Miljkovic et al. (2013) obtained the droplet population distribution and heat flux of suspended and partial wetting droplets on superhydrophobic flat surfaces by considering the coalescence-induced droplet jumping. Hu and Tang (2014) introduced the circumferential angle in the calculation of departure radius and established the dropwise condensation heat transfer model of a horizontal tube. Parin et al. (2017) compared model parameters and numerical results of Le Fevre and Rose’s model, Abu-Orabi’s model and Kim and Kim’s model, and found that the heat transfer coefficient was different, especially when the subcooling was lower. In most industrial condensers, the air in the main heat exchange zone is very little and can be negligible. The droplets on hydrophobic/superhydrophobic surfaces will be fully wetted without air. The experimental results of vertical flat surfaces (Wang et al., 2010) and horizontal tube surfaces (Hu et al., 2015) showed that the hydrophobic surfaces had a better performance of dropwise condensation heat transfer while numerical simulation on flat surfaces (Kim and Kim, 2011) and horizontal tube surfaces (Hu and Tang, 2014) both had the opposite results. In this paper, the dropwise condensation heat transfer model at pure vapour environment is built. The factors affecting heat transfer performance are investigated by numerical simulation. DOI: 10.3303/CET2081212 Paper Received: 04/06/2020; Revised: 05/07/2020; Accepted: 09/07/2020 Please cite this article as: Pang J., Gou J., Huang C., Chen K., Xiao Q., Yao S., 2020, Analysis of Factors Affecting Dropwise Condensation Heat Transfer Based on Theoretical Model, Chemical Engineering Transactions, 81, 1267-1272 DOI:10.3303/CET2081212 1267 2. Heat transfer model Consider a droplet with radius r on a plain surface with a hydrophobic coating which has fixed static contact angle θ, as shown in Figure 1, where Ts is the saturation temperature, Tsurf is the substrate temperature, and δ is the coating thickness. Figure 1: Droplet on a condensing surface with a hydrophobic coating Considering the temperature drops due to vapour-liquid interfacial resistance, curvature resistance, thermal conduction through a droplet and thermal conduction through the coating layer, the heat transfer rate through a single droplet is 2 2 (1 1 2 (1 cos / ) ) 4 sin sin min d i l c r q r r T h r        − = + + −  (1) where r is the droplet radius, rmin is the minimum radius, ΔT is the total temperature difference or subcooling temperature between the vapour and the substrate, hi is the interfacial heat transfer coefficient, θ is the contact angle, λl is the liquid thermal conductivity, δ is the coating thickness, and λc is the coating thermal conductivity. The minimum radius is expressed as: 2 s lv min fg l T r h T   =  (2) where σlv is the liquid-vapour interfacial tension, hfg is the latent heat of vaporization, and ρl is the liquid density. Actually, the subcooling temperature ΔT in the above equation is the temperature difference between the vapour and the coating surface. The minimum radius will increase as the coating conductivity resistance increases. The interfacial heat transfer coefficient is: 1/ 2 2 2 2 2 fg vc i c s s h h T TR M      =   −   (3) where αc is the condensation coefficient and generally taken as unity, M is the molar mass of water, R is the molar gas constant, and ρv is the vapour density. The droplet population distribution is: 2/ 3 max min 2 3 1 8/ 3 1/ 3 23 max min 2 3 max ( / ) ( )( exp( 3 ( ) ) ), ( ) , ( ) 1 3 ee e e e e r r r r r A r A B B r r r r r r r r A r A r ar r  −   − + + −   =    +   N (4) where parameters such as A2, A3 and so on can be referred in Kim and Kim (2011), re is the characteristic coalescence radius, rmax is the departure radius, a represents the ratio of the base area of the droplet to the radius squared, and a = when the droplet is hemispherical. When the droplet is not hemispherical, the former formula may have another form as: 2 ( ) ( ) sin r r   = N N (5) Assuming that the nucleation sites form a square array, the characteristic coalescence radius is calculated by: 1268 1/ (4 ) e s r N= (6) where Ns is the number of nucleation sites. The departure radius is obtained by the force balance between surface tension and gravity: 1/ 2 max 3 (2 3cos co 6si c ) ) n ( os s R A l lvr g             +  − − = (7) where θA is the advancing contact angle, θR is the receding contact angle, and g is the acceleration of gravity. The area ratio covered by the base area of the droplets having a radius in the range of [rmin, r] is: min min 2 2 ( ) ( sin ) ( ) , or ( ) ( sin ) ( ) r r r r r r r dr r r r dr     = =  N N (8) The dropwise condensation heat flux of the droplets having a radius in the range of [rmin, r] is: min min ( ) or ( )( ) ( ) , ( ) ( ) r r d d r r Q r Q rq r r dr q r r dr= =  N N (9) 3. Results and discussion Based on the heat transfer model established in Section 2, the influences of the number of nucleation sites, minimum radius, contact angle and departure radius are investigated numerically by a Python code. The simulation parameters selected are mainly referred to Kim and Kim (2011). 3.1 Number of nucleation sites The droplet population distribution and covered area ratio affected by the number of nucleation sites is shown in Figure 2, where the contact angle is 90 °, the advancing contact angle is 90 °, the receding contact angle is 80 °, the saturation temperature is 373 K, the wall subcooling temperature is 5 K, the coating thickness is 0.1 μm, and the coating thermal conductivity is 0.25 W m-1 K-1. The minimum radius is 4.07 nm, and the departure radius is 0.904 mm by calculation. While the number of nucleation sites varies from 109 m-2 to 1012 m-2, the characteristic coalescence radius is 15.8 μm, 5 μm, 1.58 μm and 0.5 μm. The population of small droplets increases while the number of nucleation sites increases, as shown in Figure 2a. The area occupied by the droplets having a radius in the range of [r, r+dr] is shown in Figure 2b. It can be seen that the droplets with a radius around the characteristic coalescence radius occupy more area. Based on Figure 2a, the area occupied by the small droplets increases while the number of nucleation sites increases. The area occupied by the droplets with radius in the range of [rmin, r] increases, as shown in Figure 2c. The droplets will hold 79.3 % of the surface area as the number of nucleation sites is 109 m-2, and increases to 92.6 % as the number of nucleation sites is 1012 m-2. (a) ( )rN (b) /d dr (c) ( )r Figure 2: Population distribution and area ratio occupied by droplets at different numbers of nucleation sites Heat flux affected by the number of nucleation sites is shown in Figure 3. Figure 3a shows the droplets with radius closed to the characteristic coalescence radius have a greater weight on heat transfer, and the performance of small droplets enhances while the number of nucleation sites increases. The overall heat flux is 99.92 kW m-2 as the number of nucleation sites is 109 m-2, and it increases to 620.2 kW m-2 as the number of 1269 nucleation sites is 1012 m-2. These results indicate that the number of nucleation sites impacts on the heat transfer performance very significantly. (a) /dQ dr (b) ( )Q r Figure 3: Heat flux at different numbers of nucleation sites 3.2 Minimum radius Figure 4 shows the population distribution, covered area and heat flux at different minimum radius, where the number of nucleation sites is 1010 m-2, the contact angle is 120 °, the advancing contact angle is 142 °, the receding contact angle is 102 °, the saturation temperature is 373 K, and the wall subcooling temperature is 10 K. At this case, the minimum radius calculated is 2.09 nm and set to 20.9 nm and 209 nm which are enlarged 10 times and 100 times. It can be seen in Figure 4a that due to a change of the minimum radius, the population distributions of small droplets are quite different. The droplets with a radius between the minimum radius and the characteristic coalescence radius have higher population while the minimum radius increases. As shown in Figure 4b, dQ/dr has a peak at the droplet radius of 3.4 μm, which is close to the characteristic coalescence radius of 5 μm, and much higher than 100 times of the minimum radius of 209 nm. As a result, the minimum radius affects the dropwise condensation heat transfer slightly. While increasing the minimum radius by a factor of 100, the overall heat flux declines by 2.4 %, as shown in Figure 4c. (a) ( )rN (b) /dQ dr (c) ( )Q r Figure 4: Population distribution and heat flux at different minimum radius 3.3 Contact angle Figure 5 shows the population distribution at different contact angles, where the contact angles are 90 °, 120 ° and 150 °, which are all 10 ° less than the advancing contact angle and 10 ° greater than the receding contact angle. The saturation temperature is 345 K, the wall subcooling temperature is 5 K, the coating thickness is 1 μm, the coating thermal conductivity is 0.2 W m-1 K-1, and the number of nucleation sites is 2.5×1011 m-2. In this case, the departure radius is 1.32 mm, 0.88 mm and 0.47 mm. The population distributions have small difference while calculated by ( )rN . Due to the base area of the droplets decreases, while the contact angle is increasing, the covered area ratio is 90.11 %, 68.79 % and 22.28 %, as shown in Figure 6a. The population distributions are quite different while calculated by ( )rN , and the covered area ratio is 90.11 %, 91.72 % and 89.14 %, as shown in Figure 6b. The area occupied by droplets while using ( )rN as the droplet population distribution is just the projected area while using ( )rN , which can be derived directly from Equation (5). It can be deduced that the reference Kim and Kim (2011) used ( )rN to simulate the droplet population distribution. Heat flux 1270 affected by the contact angle is shown in Figure 6c. As the contact angle increases, for the droplets with the same radius, the heat transfer resistance becomes higher, and the population distribution changes a little, so the overall heat transfer performance declines. The heat flux is 151 kW m-2， 121 kW m-2 and 57.2 kW m-2. From the experimental results of pure vapour condensation (Wang et al., 2010), the heat flux of the superhydrophobic surface is only about 40 % of that of hydrophobic surface. Figure 6c agrees well with this and is different from some other numerical results both of flat surfaces (Kim and Kim, 2011) and horizontal tube surfaces (Hu and Tang, 2014). By inference, increasing the contact angle weakens the dropwise condensation heat transfer. Heat transfer enhanced by the superhydrophobic surface is mainly caused by the reduced departure radius with the presence of non-condensable gas. (a) ( )rN (b) ( )rN Figure 5: Population distribution at different contact angles (a) ( )r using ( )rN (b) ( )r using ( )rN (c) ( )Q r using ( )rN Figure 6: Area ratio occupied by droplets and heat flux at different contact angles 3.4 Departure radius The droplet population distribution and heat flux affected by the departure radius is shown in Figure 7, where the contact angle is 90 °, the advancing contact angle is 100 °, the receding contact angle is 80 °, the saturation is 373 K, the wall subcooling temperature is 10 K, the coating thickness is 1 μm, the coating (a) ( )rN (b) ( )r (c) ( )Q r Figure 7: Population distribution and heat flux at different departure radius 1271 thermal conductivity is 0.2 W m-1 K-1, and the number of nucleation sites is 2.5×1011 m-2. The departure radius calculated is 1.28 mm, and set to 0.128 mm and 0.0128 mm to investigate its influence. As the departure radius decreases, the droplet population increases, the occupied area ratio decreases with the value of 91.08 %, 83.83 % and 67.25 %, and the overall heat flux increases obviously with the value of 350.2 kW m-2, 703.0 kW m-2 and 1,093.9 kW m-2. These results agree with the experimental results reported by Kim and Nam (2016). 4. Conclusions (1) The number of nucleation sites affects the heat transfer performance of dropwise condensation very significantly. As the number of nucleation sites increases, the number of small droplets grows, the area occupied by droplets increases and the total heat transfer performance is greatly enhanced. (2) Due to the presence of the coating conductivity resistance, the temperature difference between the vapour and the coating surface is less than that between the vapour and the substrate. The actual minimum radius will increase while the coating conductivity resistance increases. But the minimum radius affects the heat transfer performance slightly while it is much smaller than the characteristic coalescence radius. (3) As the contact angle increases, the droplet population distribution has little difference, the resistance of the droplets with the same radius increases and the overall heat transfer performance descends. It can be concluded that the heat transfer performance of the superhydrophobic surface is poorer than that of the hydrophobic surface at the pure vapour environment. (4) Dropwise condensation enhanced by the superhydrophobic surface is mainly because of the decreasing of the departure radius, with the proviso that the presence of non-condensable gas leads to droplets at suspended state. Acknowledgements The author wishes to thank the financial support from the National Natural Science Foundation of China (51706158). 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